Mesures de Monge-Ampère et mesures pluriharmoniques. (Monge-Ampère measures and pluriharmonic measures)
type: Item
instance of: scholarly article
title: Mesures de Monge-Ampère et mesures pluriharmoniques. (Monge-Ampère measures and pluriharmonic measures)
zbMATH Open document ID: 0595.32006
DOI: 10.1007/BF01161920
author: Jean-Pierre Demailly
published in: Mathematische Zeitschrift
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173853
review text: Let \(\Omega\) be a relatively compact open subset in a Stein manifold, and \(n=\dim_{{\mathbb{C}}}\Omega\). Assume that \(\Omega\) is hyperconvex, i.e. that there exists a bounded psh (plurisubharmonic) exhaustion function on \(\Omega\). A ''pluricomplex Green function'' \(u_{\Omega}\) is then naturally defined on \(\Omega \times \Omega:\) For all \(z\in \Omega\), \(u_ z(\zeta):= u_{\Omega}(z,\zeta)\) is the solution of the Dirichlet problem for the complex Monge-Ampère equation \((dd^ cu_ z)^ n=0\) on \(\Omega\) \(\setminus \{z\}\) such that \(u_ z(\zeta)=\log | \zeta - z| +O(1)\) at \(\zeta =z\); \(u_{\Omega}\) is shown to be continuous outside the diagonal and invariant under biholomorphisms. Bedford and Taylor's Monge-Ampère operators are used in conjunction with a general Lelong-Jensen formula previously found by the author [Mem. Soc. Math. Fr., Nouv. Ser. 19, 124 p. (1985; Zbl 0579.32012)] in order to construct an invariant pluricomplex Poisson kernel \(d\mu_ z(\zeta):= (2\pi)^{- n}(dd^ cu_ z(\zeta))^{n-1}\wedge d^ cu_ z(\zeta)|_{\partial \Omega},\) \((z,\zeta)\in \Omega \times \partial \Omega\). Each measure \(\mu_ z\) on \(\partial \Omega\) is such that \(\mu_ z(V)=V(z)\) for every function V pluriharmonic on \(\Omega\) and continuous on \({\bar \Omega}\); furthermore, \(\mu_ z\) is carried by the set of strictly pseudoconvex points of \(\partial \Omega\) if \({\bar \Omega}\) has a \(C^ 2\) psh defining function. The principal part of the singularity of \(d\mu_ z(\zeta)\) on the diagonal of \(\partial \Omega\) is then computed explicitly when \(\Omega\) is strictly pseudoconvex, using an osculation of \(\partial \Omega\) by balls. Through a complexification process, it is finally shown that Monge-Ampère measures provide an explicit formula representing every point of a convex compact subset \(K\subset {\mathbb{R}}^ n\) as a barycenter of the extremal points of K.
Mathematics Subject Classification ID: 32A25
Mathematics Subject Classification ID: 32C30
Mathematics Subject Classification ID: 32F45
Mathematics Subject Classification ID: 32U05
Mathematics Subject Classification ID: 31C10
Mathematics Subject Classification ID: 32E10
zbMATH DE Number: 3957595
zbMATH Keywords: pluricomplex Green function
zbMATH Keywords: Lelong-Jensen formula
zbMATH Keywords: Monge-Ampère measures
zbMATH Keywords: plurisubharmonic exhaustion function
zbMATH Keywords: hyperconvex domain
zbMATH Keywords: pluriharmonic measures
zbMATH Keywords: Choquet's theorem
zbMATH Keywords: barycentric representation
MaRDI profile type: Publication
cites work: A new capacity for plurisubharmonic functions
cites work: Pseudoconvex domains: bounded strictly plurisubharmonic exhaustion functions
cites work: Embedding Strictly Pseudoconvex Domains in Convex Domains
cites work: Subharmonicity for uniform algebras
cites work: Extremal plurisubharmonic functions and invariant pseudodistances
cites work: La métrique de Kobayashi et la représentation des domaines sur la boule
cites work: Solving the degenerate complex Monge-Ampère equation with one concentrated singularity
cites work: Q5508589
cites work: Stetige streng pseudokonvexe Funktionen
cites work: On Some Extremal Functions and their Applications in the Theory of Analytic Functions of Several Complex Variables
cites work: Extremal plurisubharmonic functions in $C^N$
cites work: Q3701749
cites work: Q5542428
OpenAlex ID: W2006265208
description: scientific article; zbMATH DE number 3957595